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Communication bandwidth and network topology are two important factors that affect performance of distributed consensus for multi-agent systems. To deal with the limited bandwidth problem, quantization schemes have been presented. The available works about quantized average consensus require the agents to communicate in balanced digraphs, which is difficult to realize in the real world. By dropping this stringent requirement, we revisit an existing average consensus protocol based on logarithmic quantization scheme and investigate the quantized consensus problem on general strongly connected directed networks. An upper bound of the quantization precision parameter is derived to design the suitable logarithmic quantizer. Based on matrix transformation and Lyapunov stability theory, we provide a testable condition under which the weighted average consensus can be achieved with the proposed quantized protocol.